Abstract

Blazed diffraction gratings are of enormous practical importance for imaging and analyzing hybrid optical systems. The intermediate diffraction regime is characterized by the transition from the scalar to the rigorous electromagnetic theory. An effect known as shadowing occurs and reduces the diffraction efficiency. Based on rigorous calculations for optimized sawtooth-shaped and binary-multilevel blaze profiles, we deduce a semianalytical model describing the shadowing phenomenon for the general case of oblique incidence. We discuss illumination both from air and from the substrate. Though a multilevel blaze possesses a discrete substructure, our shadowing model remains valid, if substructural effects are neglected. We find that electromagnetic effects due to the passive blaze facet lead to the efficiency reduction, and the blazing efficiency shows a linear dependence on the ratio of blaze wavelength to grating period. Our shadowing model is applied to predict the performance of a Littrow-like blazing condition in transmission geometry as, e.g., for a diffractive solid immersion lens.

© 2008 Optical Society of America

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2007 (1)

O. Sandfuchs, A. Pesch, and R. Brunner, “Rigorous modeling of dielectric and metallic blaze gratings in the intermediate structure regime,” Proc. SPIE 6675, 66750I-1-8 (2007).

2006 (2)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, “Rigorous analysis of shadowing effects in blazed transmission gratings,” Opt. Lett. 31, 3638-3640 (2006).
[CrossRef] [PubMed]

2004 (3)

R. Brunner, M. Burkhardt, A. Pesch, O. Sandfuchs, M. Ferstl, S. Hohng, and J. O. White, “Diffraction-based solid immersion lens,” J. Opt. Soc. Am. A 21, 1186-1191 (2004).
[CrossRef]

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: Simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

2003 (3)

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, “Diffractive-refractive hybrid microscope objective for 193 nm inspection systems,” Proc. SPIE 5177, 9-15 (2003).
[CrossRef]

R. Brunner, R. Steiner, H.-J. Dobschal, D. Martin, M. Burkhardt, and M. Helgert, “New solutions to realize complex optical systems by a combination of diffractive and refractive optical components,” Proc. SPIE 5183, 47-55 (2003).
[CrossRef]

P.-K. Wei, H.-L. Chou, and W.-L. Chang, “Diffraction-induced near-field optical images in mesoscale air-dielectric structures,” J. Opt. Soc. Am. B 20, 1503-1507 (2003).
[CrossRef]

2002 (1)

M. S. L. Lee, Ph. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4, S119-S124 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (2)

1998 (1)

1997 (3)

1996 (2)

T. Delort, D. Maystre, and J. P. Laude, “Perfect blazing for transmission gratings: Generalized and numerical verification,” J. Opt. Soc. Am. A 13, 2034-2040 (1996).
[CrossRef]

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile. Theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

1995 (2)

1994 (1)

1993 (1)

1991 (1)

G. J. Swanson, “Binary optics technology: Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Tech. Rep. 914, Massachusetts Institute of Technology Lincoln Laboratory, Cambridge, Massachusetts, 1991.

1990 (1)

S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615-2616 (1990).
[CrossRef]

1983 (1)

S.-Y. Kim, J.-W. Ra, and S.-Y. Shin, “Edge diffraction by dielectric wedge of arbitrary angle,” Electron. Lett. 19, 851-853 (1983).
[CrossRef]

Aagedal, H.

Astilean, S.

Barton, I. M.

Blough, C. G.

Blough, G.

W. Knapp, G. Blough, K. Khajurival, R. Michaels, B. Tatian, and B. Volk, “Optical design comparison of 60° eyepieces: One with a diffractive surface and one with aspherics,” Appl. Opt. 34, 4756-4760 (1997).
[CrossRef]

Britten, J. A.

Brunner, R.

O. Sandfuchs, A. Pesch, and R. Brunner, “Rigorous modeling of dielectric and metallic blaze gratings in the intermediate structure regime,” Proc. SPIE 6675, 66750I-1-8 (2007).

O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, “Rigorous analysis of shadowing effects in blazed transmission gratings,” Opt. Lett. 31, 3638-3640 (2006).
[CrossRef] [PubMed]

R. Brunner, M. Burkhardt, A. Pesch, O. Sandfuchs, M. Ferstl, S. Hohng, and J. O. White, “Diffraction-based solid immersion lens,” J. Opt. Soc. Am. A 21, 1186-1191 (2004).
[CrossRef]

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

R. Brunner, R. Steiner, H.-J. Dobschal, D. Martin, M. Burkhardt, and M. Helgert, “New solutions to realize complex optical systems by a combination of diffractive and refractive optical components,” Proc. SPIE 5183, 47-55 (2003).
[CrossRef]

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, “Diffractive-refractive hybrid microscope objective for 193 nm inspection systems,” Proc. SPIE 5177, 9-15 (2003).
[CrossRef]

Buchda, G.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

Burkhardt, M.

R. Brunner, M. Burkhardt, A. Pesch, O. Sandfuchs, M. Ferstl, S. Hohng, and J. O. White, “Diffraction-based solid immersion lens,” J. Opt. Soc. Am. A 21, 1186-1191 (2004).
[CrossRef]

R. Brunner, R. Steiner, H.-J. Dobschal, D. Martin, M. Burkhardt, and M. Helgert, “New solutions to realize complex optical systems by a combination of diffractive and refractive optical components,” Proc. SPIE 5183, 47-55 (2003).
[CrossRef]

Cambril, E.

M. S. L. Lee, Ph. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4, S119-S124 (2002).
[CrossRef]

Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A 16, 1143-1156 (1999).
[CrossRef]

Chang, W.-L.

Chavel, P.

M. S. L. Lee, Ph. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4, S119-S124 (2002).
[CrossRef]

Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A 16, 1143-1156 (1999).
[CrossRef]

Chen, Y.

M. S. L. Lee, Ph. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4, S119-S124 (2002).
[CrossRef]

Chou, H.-L.

Delort, T.

Dixit, S. N.

Dobschal, H.-J.

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, “Diffractive-refractive hybrid microscope objective for 193 nm inspection systems,” Proc. SPIE 5177, 9-15 (2003).
[CrossRef]

R. Brunner, R. Steiner, H.-J. Dobschal, D. Martin, M. Burkhardt, and M. Helgert, “New solutions to realize complex optical systems by a combination of diffractive and refractive optical components,” Proc. SPIE 5183, 47-55 (2003).
[CrossRef]

Drauschke, A.

Ferstl, M.

Fleming, M. B.

Gale, M. T.

Golub, M.

Grann, E. B.

Helgert, M.

R. Brunner, R. Steiner, H.-J. Dobschal, D. Martin, M. Burkhardt, and M. Helgert, “New solutions to realize complex optical systems by a combination of diffractive and refractive optical components,” Proc. SPIE 5183, 47-55 (2003).
[CrossRef]

Hessler, T.

Hohng, S.

Horn, U.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

Hutley, M. C.

Hyde, R. A.

Khajurival, K.

W. Knapp, G. Blough, K. Khajurival, R. Michaels, B. Tatian, and B. Volk, “Optical design comparison of 60° eyepieces: One with a diffractive surface and one with aspherics,” Appl. Opt. 34, 4756-4760 (1997).
[CrossRef]

Kim, S.-Y.

S.-Y. Kim, J.-W. Ra, and S.-Y. Shin, “Edge diffraction by dielectric wedge of arbitrary angle,” Electron. Lett. 19, 851-853 (1983).
[CrossRef]

Kino, G. S.

S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615-2616 (1990).
[CrossRef]

Kleemann, B. H.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile. Theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Knapp, W.

W. Knapp, G. Blough, K. Khajurival, R. Michaels, B. Tatian, and B. Volk, “Optical design comparison of 60° eyepieces: One with a diffractive surface and one with aspherics,” Appl. Opt. 34, 4756-4760 (1997).
[CrossRef]

Kunz, R. E.

Lalanne, Ph.

M. S. L. Lee, Ph. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4, S119-S124 (2002).
[CrossRef]

Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A 16, 1143-1156 (1999).
[CrossRef]

Laude, J. P.

Launois, H.

Lee, M. S. L.

M. S. L. Lee, Ph. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4, S119-S124 (2002).
[CrossRef]

Levy, U.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: Simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Lu., K.

Mack, S. K.

Mansfield, S. M.

S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615-2616 (1990).
[CrossRef]

Marom, E.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: Simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Martin, D.

R. Brunner, R. Steiner, H.-J. Dobschal, D. Martin, M. Burkhardt, and M. Helgert, “New solutions to realize complex optical systems by a combination of diffractive and refractive optical components,” Proc. SPIE 5183, 47-55 (2003).
[CrossRef]

Maystre, D.

Menck, A.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

Mendlovic, D.

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: Simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Michaels, R.

W. Knapp, G. Blough, K. Khajurival, R. Michaels, B. Tatian, and B. Volk, “Optical design comparison of 60° eyepieces: One with a diffractive surface and one with aspherics,” Appl. Opt. 34, 4756-4760 (1997).
[CrossRef]

Michaels, R. L.

Missig, M. D.

Mitreiter, A.

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile. Theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Noponen, E.

Pätz, D.

Perry, M. D.

Pesch, A.

O. Sandfuchs, A. Pesch, and R. Brunner, “Rigorous modeling of dielectric and metallic blaze gratings in the intermediate structure regime,” Proc. SPIE 6675, 66750I-1-8 (2007).

R. Brunner, M. Burkhardt, A. Pesch, O. Sandfuchs, M. Ferstl, S. Hohng, and J. O. White, “Diffraction-based solid immersion lens,” J. Opt. Soc. Am. A 21, 1186-1191 (2004).
[CrossRef]

Pfeil, A. v.

Pommet, D. A.

Ra, J.-W.

S.-Y. Kim, J.-W. Ra, and S.-Y. Shin, “Edge diffraction by dielectric wedge of arbitrary angle,” Electron. Lett. 19, 851-853 (1983).
[CrossRef]

Rathsfeld, A.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

Rodier, J. C.

M. S. L. Lee, Ph. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4, S119-S124 (2002).
[CrossRef]

Rossi, M.

Rudolf, K.

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, “Diffractive-refractive hybrid microscope objective for 193 nm inspection systems,” Proc. SPIE 5177, 9-15 (2003).
[CrossRef]

Ruoff, J.

Rushford, M. C.

Sandfuchs, O.

Schmidt, G.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

Shin, S.-Y.

S.-Y. Kim, J.-W. Ra, and S.-Y. Shin, “Edge diffraction by dielectric wedge of arbitrary angle,” Electron. Lett. 19, 851-853 (1983).
[CrossRef]

Sinzinger, S.

Steiner, R.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

R. Brunner, R. Steiner, H.-J. Dobschal, D. Martin, M. Burkhardt, and M. Helgert, “New solutions to realize complex optical systems by a combination of diffractive and refractive optical components,” Proc. SPIE 5183, 47-55 (2003).
[CrossRef]

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, “Diffractive-refractive hybrid microscope objective for 193 nm inspection systems,” Proc. SPIE 5177, 9-15 (2003).
[CrossRef]

Summers, L. J.

Swanson, G. J.

G. J. Swanson, “Binary optics technology: Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Tech. Rep. 914, Massachusetts Institute of Technology Lincoln Laboratory, Cambridge, Massachusetts, 1991.

Tatian, B.

W. Knapp, G. Blough, K. Khajurival, R. Michaels, B. Tatian, and B. Volk, “Optical design comparison of 60° eyepieces: One with a diffractive surface and one with aspherics,” Appl. Opt. 34, 4756-4760 (1997).
[CrossRef]

Testorf, M.

Thomas, I. M.

Turunen, J.

Vasara, A.

Volk, B.

W. Knapp, G. Blough, K. Khajurival, R. Michaels, B. Tatian, and B. Volk, “Optical design comparison of 60° eyepieces: One with a diffractive surface and one with aspherics,” Appl. Opt. 34, 4756-4760 (1997).
[CrossRef]

Wei, P.-K.

Weissenberg, S.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

White, J. O.

Wyrowski, F.

A. v. Pfeil, F. Wyrowski, A. Drauschke, and H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304-3313 (2000).
[CrossRef]

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile. Theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Zibold, A. M.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. Lett. (1)

S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615-2616 (1990).
[CrossRef]

Comm. Comp. Phys. (1)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

Electron. Lett. (1)

S.-Y. Kim, J.-W. Ra, and S.-Y. Shin, “Edge diffraction by dielectric wedge of arbitrary angle,” Electron. Lett. 19, 851-853 (1983).
[CrossRef]

J. Mod. Opt. (1)

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile. Theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

M. S. L. Lee, Ph. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4, S119-S124 (2002).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

U. Levy, E. Marom, and D. Mendlovic, “Thin element approximation for the analysis of blazed gratings: Simplified model and validity limits,” Opt. Commun. 229, 11-21 (2004).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (4)

O. Sandfuchs, A. Pesch, and R. Brunner, “Rigorous modeling of dielectric and metallic blaze gratings in the intermediate structure regime,” Proc. SPIE 6675, 66750I-1-8 (2007).

R. Brunner, R. Steiner, H.-J. Dobschal, D. Martin, M. Burkhardt, and M. Helgert, “New solutions to realize complex optical systems by a combination of diffractive and refractive optical components,” Proc. SPIE 5183, 47-55 (2003).
[CrossRef]

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, “Diffractive-refractive hybrid microscope objective for 193 nm inspection systems,” Proc. SPIE 5177, 9-15 (2003).
[CrossRef]

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE 5567, 887-893 (2004).
[CrossRef]

Other (1)

G. J. Swanson, “Binary optics technology: Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Tech. Rep. 914, Massachusetts Institute of Technology Lincoln Laboratory, Cambridge, Massachusetts, 1991.

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Figures (10)

Fig. 1
Fig. 1

Different profile geometries: Sawtooth-shaped (solid curve), binary-multilevel (dashed curve), and holographic blaze structure (dotted curve).

Fig. 2
Fig. 2

Schematic drawings of various illumination conditions for a blaze profile resulting in different shadowing zones (shaded areas): (a) Direct geometrical shadow, (b) dead blaze zone (right-shaded and left-shaded), and (c) no ray-optical shadow at all.

Fig. 3
Fig. 3

Efficiency behavior as a function of the scaled grating period: (a) for normal incidence ( θ = 0 ) and various refractive indices n, and (b) for n = 1.49 comparing rigorous simulations (solid curves) with EST (dashed curves) in the low diffraction orders m = 1 , 2, and 3.

Fig. 4
Fig. 4

Optimum angle of inclination β of the passive blaze facet as a function of the scaled grating period for various incidence angles θ (solid curves). Dashed curves for comparison; β = 90 ° θ = const. and β = 90 ° θ 1 ( g / λ B ) , respectively.

Fig. 5
Fig. 5

Efficiency behavior as a function of the scaled grating period for three different incidence angles θ in degrees, where m = 1 and n = 1.49 .

Fig. 6
Fig. 6

Dependence of the shadowing strength c 1 on the incidence angle θ in the first diffraction order: (a) for n = 1.49 and different choices of β, and (b) for β = β opt from rigorous calculations and the model fit based on Eq. (12).

Fig. 7
Fig. 7

Schematic of the two different illumination conditions: (a) from the air and (b) from the substrate.

Fig. 8
Fig. 8

Comparison of the shadowing strength c m ( n , θ = 0 ° ) for the two cases of illumination shown in Fig. 7.

Fig. 9
Fig. 9

Efficiency reduction for multilevel blaze structure as a function of scaled grating period for the three states of polarization (solid curves) compared to the ideal sawtooth-shaped blaze structure (dotted curves).

Fig. 10
Fig. 10

Comparison of the diffraction efficiency of a dSIL as function of incidence angles θ using the three theoretical models: Rigorous calculations, the electromagnetic model [Eq. (17)], and the EST [Eq. (5)].

Tables (1)

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Table 1 Parameters of Eq. (12) for the Shadowing Coefficient c + 1 ( n , θ ) for UP Light at Oblique Incidence with the Angle θ Taken in Degrees

Equations (19)

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η = η max Δ η = η max ( 1 Δ η η max ) ,
η max TE = 4 n 0 n cos θ cos θ m ( n 0 cos θ + n cos θ m ) 2 ,
η max TM = 4 n 0 n cos θ cos θ m ( n cos θ + n 0 cos θ m ) 2 ,
η max UP = 1 2 ( η max TE + η max TM ) ,
Δ η η max = tan ( θ m ) n n 0 m λ B g .
Δ η η max = m 2 ( n n 0 ) n 0 ( λ B g ) 2 + ,
Δ η η max = sin ( θ m θ ) n n 0 cos ( θ m θ ) m λ B g cos θ .
Δ η η max = m 2 n ( n n 0 ) ( λ B g ) 2 + .
η rig , θ = 0 = η max [ 1 c m ( n ) ( λ B g ) ν ] ,
tan α opt = m λ B g n cos θ m n 0 cos θ .
cos β opt sin θ m = 1 n ( n 0 sin θ + m λ B g ) .
h opt = g sin α opt sin β opt sin ( α opt + β opt ) ,
η rig = η max [ 1 c m ( n , θ ) ( λ B g ) ν ] ,
c + 1 ( n , θ ) c + 1 ( n , θ = 0 ) a ( n ) θ + b ( n ) θ 2 .
c + 1 ( n , θ = 0 ) 0.76 [ n ( n 1 ) ] 0.44 .
h N opt = N 1 N h blaze opt = N 1 N g tan α opt .
θ m = + 1 = θ ,
g θ = λ B ( n 1 ) sin θ .
η dSIL ( n , θ ) = η max UP ( n , θ ) [ 1 c + 1 ( n , θ ) ( n 1 ) sin θ ] .

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